proposed differential equation for spin 12

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1、1IntroductionIntrinsicspinandorbitalangularmomentumarefundamentallythesame.Onedifferencebetweenthemappearstobethatwhileorbitalangularmomen-tumhasawave-mechanicaldescriptionthroughLegendre’sequation,spindoesnot.However,inthispaper,weproposeadifferentialeigenvalueequa-

2、tionforspin1/2.TheargumentsleadingtoourproposedequationarebasedontheLand´einterpretationofquantummechanics[1-4].Inthisinterpretation,theprobability-amplitudecharacterofaneigenfunctionorwavefunctionisstressed.Suchaprobabilityamplitudeconnectswell-definedinitialandfina

3、lstates[5].Per-hapsthemostimportantsuchprobabilityamplitudesresultfromsolutionofthetime-independentSchr¨odingerequation2h¯2−∇ψ(r)+V(r)ψ(r)=Eψ(r).(1)2mTheeigenfunctionψ(r)connectsaninitialstatedefinedbytheeigenvalueEandafinalstatedefinedbytheeigenvaluer.Toemphasizethei

4、nitialandfinalstate,wewritetheeigenfunctionasψ(r)=ψ(En;r),sothattheequationreads2h¯2−∇ψ(En;r)+V(r)ψ(En;r)=Enψ(En;r).(2)2mTheSchr¨odingerequationis,ofcourse,atypicaleigenvalueequation.An-otherexampleistheLegendreequationarXiv:quant-ph/0411060v19Nov2004L2(θ,ϕ)Y(θ,ϕ)=l

5、(l+1)¯h2Y(θ,ϕ).(3)lmlm2Inthisequation,theinitialstatecorrespondstotheeigenvaluesl(l+1)¯handmh¯whilethefinalstatecorrespondstotheangularposition(θ,ϕ)[6].Forthisreason,wemaywritetheeigenfunctionsasYlm(θ,ϕ)=Y(l,m;θ,ϕ).WenoticethatinboththeSchr¨odingerandtheLegendreequa

6、tions,theoperatoriscastintermsofthevariableswhichalsodefinethefinaleigenvalue.Wetakethistobeageneralfeature,whichweexploitinoursearchforadifferentialequationforspin.12SpinEigenvectorsAsconventionallytreated,spinissuchthatboththeinitialandfinalstatescorrespondtodiscrete

7、eigenvalues.Adifferentialeigenvalueequationrequiresacontinuousvariablefortheoperatortoacton.But,thePaulispinvectors!!+1−0[ξz,z]=and[ξz,z]=(4)01donotappeartoofferacontinuousvariablewhichmightmakepossibleadifferentialtreatmentofspin.However,thePaulivectorsarespecialised

8、formsofthegeneralizedspineigenvectors[5]!χ((+1)(ba);(+1)(bb))cosθ/2cosθ′/2+ei(ϕ−ϕ′)sinθ/2sinθ′/2[ξ+]=22=ba1(ba)1(bb)′i(ϕ−ϕ′)′,bbχ((+);(−))cos